Okay, I hope I answer your questions...
It says to wash with a non-medicated soap. I would recommend that you do not use a bar soap. They can be very drying to your skin, and the
Benzamycin will be plenty drying. I would recommend that you use a non-soap cleanser (perhaps Cetaphil, or Purpose by Johnson & Johnson).
If your face does feel greasy during the day, I do not think you are causing it harm by splashing with some water in between your morning and evening washing. However, make sure your hands and your towels are clean! If you touch your face too much, you could increase your chance of breaking out.
Another option is oil absorbing papers. They can be used to blot excess oil in the middle of the day. I know Clean and Clear makes some, so does the Body Shop, and Palladio. They are pretty inexpensive.
Now, your moisturizer question. Try washing your face and using the
Benzamycin for a few days without a moisturizer. If you find your face gets dry, I would use a moisturizer. After you wash your face (with your soap-free cleanser), apply an oil-free moisturizer (I use Oil of Olay's Oil Free Active Hydrating...). Let it "soak" into your skin, then apply your
Benzamycin.
I hope that I have helped. Good luck!